Elliptic curve 135.3-b1 over number field Q(10)\Q(\sqrt{10})

• Label: 2.2.40.1-135.3-b1
• Base field: $\Q(\sqrt{10})$
• Conductor norm: $135$
• CM: no
• Base change: no
• Q-curve: no
• Torsion order: $1$
• Rank: $1$

Base field $\Q(\sqrt{10})$

Generator $a$, with minimal polynomial $x^{2} - 10$; class number $2$.

Weierstrass equation

${y}^2+a{x}{y}+a{y}={x}^{3}-a{x}^{2}+\left(4a-3\right){x}+5a+18$

This is not a global minimal model: it is minimal at all primes except $(2,a)$. No global minimal model exists.

Mordell-Weil group structure

$\Z$

Mordell-Weil generators

$P$	$\hat{h}(P)$	Order
$\left(1 : 4 : 1\right)$	$0.51436368130789735042071775596677115710$	$\infty$

Invariants

Conductor:	$\frak{N}$	=	$(-4a-5)$	=	$(3,a+2)^{3}\cdot(5,a)$
Conductor norm:	$N(\frak{N})$	=	$135$	=	$3^{3}\cdot5$
Discriminant:	$\Delta$	=	$-32000a-40000$
Discriminant ideal:	$(\Delta)$	=	$(-32000a-40000)$	=	$(2,a)^{12}\cdot(3,a+2)^{3}\cdot(5,a)^{7}$
Discriminant norm:	$N(\Delta)$	=	$8640000000$	=	$2^{12}\cdot3^{3}\cdot5^{7}$
Minimal discriminant:	$\frak{D}_{\mathrm{min}}$	=	$(-500a-625)$	=	$(3,a+2)^{3}\cdot(5,a)^{7}$
Minimal discriminant norm:	$N(\frak{D}_{\mathrm{min}})$	=	$2109375$	=	$3^{3}\cdot5^{7}$
j-invariant:	$j$	=	$\frac{10728071}{625} a - \frac{6736561}{125}$
Endomorphism ring:	$\mathrm{End}(E)$	=	$\Z$
Geometric endomorphism ring:	$\mathrm{End}(E_{\overline{\Q}})$	=	$\Z$    (no potential complex multiplication)
Sato-Tate group:	$\mathrm{ST}(E)$	=	$\mathrm{SU}(2)$

BSD invariants

Analytic rank:	$r_{\mathrm{an}}$	=	$1$
Mordell-Weil rank:	$r$	=	$1$
Regulator:	$\mathrm{Reg}(E/K)$	≈	$0.51436368130789735042071775596677115710$
Néron-Tate Regulator:	$\mathrm{Reg}_{\mathrm{NT}}(E/K)$	≈	$1.02872736261579470084143551193354231420$
Global period:	$\Omega(E/K)$	≈	$16.113616945928583209617692822213419640$
Tamagawa product:	$\prod_{\frak{p}}c_{\frak{p}}$	=	$1$  =  $1\cdot1\cdot1$
Torsion order:	$\#E(K)_{\mathrm{tor}}$	=	$1$
Special value:	$L^{(r)}(E/K,1)/r!$	≈	$2.6209777325662875240266073791396135238$
Analytic order of Ш:	Ш${}_{\mathrm{an}}$	=	$1$ (rounded)

BSD formula

$\displaystyle 2.620977733 \approx L'(E/K,1) \overset{?}{=} \frac{ \# Ш(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \approx \frac{ 1 \cdot 16.113617 \cdot 1.028727 \cdot 1 } { {1^2 \cdot 6.324555} } \approx 2.620977733$

Local data at primes of bad reduction

This elliptic curve is not semistable. There are 2 primes $\frak{p}$ of bad reduction. Primes of good reduction for the curve but which divide the discriminant of the model above (if any) are included.

$\mathfrak{p}$	$N(\mathfrak{p})$	Tamagawa number	Kodaira symbol	Reduction type	Root number	$\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}$)	$\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}$)	$\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))$
$(2,a)$	$2$	$1$	$I_0$	Good	$1$	$0$	$0$	$0$
$(3,a+2)$	$3$	$1$	$II$	Additive	$-1$	$3$	$3$	$0$
$(5,a)$	$5$	$1$	$I_{7}$	Non-split multiplicative	$1$	$1$	$7$	$7$

Galois Representations

The mod $p$ Galois Representation has maximal image for all primes $p < 1000$ .

Isogenies and isogeny class

This curve has no rational isogenies. Its isogeny class 135.3-b consists of this curve only.

Base change

This elliptic curve is not a $\Q$-curve.

It is not the base change of an elliptic curve defined over any subfield.